The Ptolemy field of $3$–manifold representations

Abstract

The Ptolemy coordinates for boundary-unipotent $SL\left(n,ℂ\right)$–representations of a $3$–manifold group were introduced by Garoufalidis, Thurston and Zickert [arXiv:1111.2828] inspired by the $\mathcal{A}$–coordinates on higher Teichmüller space due to Fock and Goncharov. We define the Ptolemy field of a (generic) $PSL\left(2,ℂ\right)$-representation and prove that it coincides with the trace field of the representation. This gives an efficient algorithm to compute the trace field of a cusped hyperbolic manifold.